theorem Th66: :: MESFUN12:66

for X1, X2 being non empty set
for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL
for E1, E2 being Element of sigma (measurable_rectangles (S1,S2)) st E1 = dom f & f is nonnegative & f is E1 -measurable holds
( Integral1 (M1,f) is nonnegative & Integral1 (M1,(f | E2)) is nonnegative & Integral2 (M2,f) is nonnegative & Integral2 (M2,(f | E2)) is nonnegative )